I'm not sure this is actually going to be a useful explanation, but I'll give it a go.
One problem has to do with rotational symmetry. The selection of a juggling pin is perhaps not the best example, but let us assume for now that the juggling pin has a dot somewhere on its surface that ensures it is not rotationally symmetric about any axis. If you are only concerned with the with orienting a body subject to symmetry you can sometimes do better (for example, a perfectly uniform sphere would only require three components), but the general case requires six components.
Your approach of pinning two atoms in the pin effectively specifies an axis. Yes, it would accurately position and constrain two points on the pin. Those two points form a line. Consider what happens if I now rotate the rest of the atoms in the pin around that line. I have not changed the representation in your coordinate system, because I've kept the two atoms you specified fixed, but I have changed the orientation of the pin for an external observer. If, on the other hand, you constrained another atom that was not colinear with the first two, I would no longer be allowed this freedom of rotation. The system would be said to be fully constrained.
To put this into concrete terms with our pin, let's assume the two atoms I picked were the center of the top and the center of the bottom. Because the pin is symmetric about this axis, with five numbers I can specify the orientation of the pin sufficiently to juggle it, but what about the dot I mentioned before? By rotating the pin about this principal axis I could change the position of the dot without changing the coordinates of the pin.
Stated more simply: if I tell you the position of the center of the earth and my location in terms of latitude and longitude, you can locate me precisely, but you can't tell whether I'm looking north or south.
Also, you can only encode a (countably infinite) subset of the reals by interleaving digits. Consider the set of irrational numbers. Any scheme you provide that tries to reduce the dimension of a vector of n reals into n-1 will face the same problem: you cannot project from a space who's components are made up of members uncountably infinite sets into a space of lower dimension without losing information.
One problem has to do with rotational symmetry. The selection of a juggling pin is perhaps not the best example, but let us assume for now that the juggling pin has a dot somewhere on its surface that ensures it is not rotationally symmetric about any axis. If you are only concerned with the with orienting a body subject to symmetry you can sometimes do better (for example, a perfectly uniform sphere would only require three components), but the general case requires six components.
Your approach of pinning two atoms in the pin effectively specifies an axis. Yes, it would accurately position and constrain two points on the pin. Those two points form a line. Consider what happens if I now rotate the rest of the atoms in the pin around that line. I have not changed the representation in your coordinate system, because I've kept the two atoms you specified fixed, but I have changed the orientation of the pin for an external observer. If, on the other hand, you constrained another atom that was not colinear with the first two, I would no longer be allowed this freedom of rotation. The system would be said to be fully constrained.
To put this into concrete terms with our pin, let's assume the two atoms I picked were the center of the top and the center of the bottom. Because the pin is symmetric about this axis, with five numbers I can specify the orientation of the pin sufficiently to juggle it, but what about the dot I mentioned before? By rotating the pin about this principal axis I could change the position of the dot without changing the coordinates of the pin.
Stated more simply: if I tell you the position of the center of the earth and my location in terms of latitude and longitude, you can locate me precisely, but you can't tell whether I'm looking north or south.
Also, you can only encode a (countably infinite) subset of the reals by interleaving digits. Consider the set of irrational numbers. Any scheme you provide that tries to reduce the dimension of a vector of n reals into n-1 will face the same problem: you cannot project from a space who's components are made up of members uncountably infinite sets into a space of lower dimension without losing information.